1. The first question involves evaluating the expression $(5 \times 2) \times 3 = 30$. This is a simple arithmetic calculation. 2. Next, the function $f(x)$ is defined piecewise as: $$ f(x) = \begin{cases} 0 & \text{for } -\pi < x < 0 \\ 1 & \text{for } 0 < x < \pi \end{cases} $$ This is a step function with values 0 and 1 in the given intervals. 3. The partial derivatives of the function $f(x,y) = x^2 y - x y^2$ are requested: - Partial derivative with respect to $x$: $$ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2 y - x y^2) = 2 x y - y^2 $$ - Partial derivative with respect to $y$: $$ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2 y - x y^2) = x^2 - 2 x y $$ 4. The differential equations given are: - $$\frac{dy}{dx} = \frac{x^2 + y^2}{2 x^2}$$ - $$\frac{dy}{dx} = \frac{x + y}{x}$$ These are expressions for the derivative $\frac{dy}{dx}$ in terms of $x$ and $y$. 5. The function $z = a x + b y + a^2 + b^2$ is given, which is a function of variables $a$, $b$, $x$, and $y$. 6. The series to test is: $$ 1 + \frac{x}{2} + \frac{x^2}{3^2} + \frac{x^3}{4^3} + \cdots $$ for all positive values of $x$. Since the user asked to fix the questions, the first question is the arithmetic expression. We confirm it is correct: Step 1: Calculate $5 \times 2 = 10$ Step 2: Multiply the result by 3: $10 \times 3 = 30$ Therefore, the expression $(5 \times 2) \times 3 = 30$ is correct. Since the user asked to fix the questions, the first question is already correct and needs no fixing. "slug": "arithmetic evaluation", "subject": "algebra", "desmos": {"latex": "y=x", "features": {"intercepts": true, "extrema": true}}, "q_count": 6